Avaroth Harindranath, Saha Institute of Nuclear Physics, Calcutta, India

Dipankar Chakrabarty, Florida State University

Lubo Martinovic, Institute of Physics Institute, Bratislava, Slovakia

Grigorii Pivovarov, Institute for Nuclear Research, Moscow, Russia

Peter Peroncik, Richard Lloyd, John R. Spence, James P. Vary,

Iowa State University

Coherent States and Spontaneous Symmetry Breaking in Light Front Scalar Field Theory

I. Ab initio approach to quantum many-body systemsII. Constituent quark models & light front 4

1+1

III. Conclusions and Outlook

LC2005 - Cairns, AustraliaJuly 7-15 , 2005

Constructing the non-perturbative theory bridge between“Short distance physics” “Long distance physics”

Asymptotically free current quarks Constituent quarksChiral symmetry Broken Chiral symmetryHigh momentum transfer processes Meson and Baryon Spectroscopy

NN interactions

H(bare operators) HeffBare transition operators Effective charges, transition ops, etc.

Bare NN, NNN interactions Effective NN, NNN interactions fitting 2-body data describing low energy nuclear dataShort range correlations & Mean field, pairing, & strong tensor correlations quadrupole, etc., correlations

BOLD CLAIM We now have the tools to accomplish this program in

nuclear many-body theory

H acts in its full infinite Hilbert Space

Heff of finite subspace

Ab Initio Many-Body Theory

€

H =Trel + V (a)

€

HA = Trel +V = [(

r p i −

r p j )

2

2mA+ Vij

i< j

A

∑ ] + VNNN

P. Navratil, J.P. Vary and B.R. Barrett, Phys. Rev. Lett. 84, 5728(2000); Phys. Rev. C62, 054311(2000)C. Viazminsky and J.P. Vary, J. Math. Phys. 42, 2055 (2001);K. Suzuki and S.Y. Lee, Progr. Theor. Phys. 64, 2091(1980);

K. Suzuki, ibid, 68, 246(1982); K. Suzuki and R. Okamoto, ibid, 70, 439(1983)

Preserves the symmetries of the full Hamiltonian:Rotational, translational, parity, etc., invariance

Effective Hamiltonian for A-ParticlesLee-Suzuki-Okamoto Method plus Cluster Decomposition

Select a finite oscillator basis space (P-space) and evaluate an - body cluster effective Hamiltonian:

Guaranteed to provide exact answers as or as .

€

a

€

a → A

€

P → 1

NMIN=0

NMAX=6configuration

“6h” configuration for 6Li

Select a subsystem (cluster) of a < A Fermions.

Develop a unitary transformation for a finite space,the “P-space” that generates the exact low-lying spectraof that cluster subsytem. Since it is unitary, it preservesall symmetries.

Construct the A-Fermion Hamiltonian from this a-Fermioncluster Hamiltonian and solve for the A-Fermion spectraby diagonalization.

Guaranteed to provide the full spectra as either a --> A or as P --> 1

Guide to the methodology

€

H (a) =(Pa +ωTω)−1/ 2(Pa + PaωTQa )Ha

Ω(QaωPa + Pa )(Pa + ωTω)−1/ 2

HaΩ k =Ek k

αQ ω αP = αQ kk∈K∑ ˆ k αP

where: ˆ k αP =Inverse{k αP }

Pa = αPP∈P∑ αP

Qa = αQQ∈Q∑ αQ

Pa +Qa ≈1a

Key equations to solve at the a-body cluster level

Solve a cluster eigenvalue problem in a very large but finite basisand retain all the symmetries of the bare Hamiltonian

Historical PerspectiveNuclei with Realistic Interactions

1985 - Exact solution of Fadeev equations (3-Fermions)1991 - Exact solution of Fadeev-Jacobovsky equations (4-Fermions)2000 - Green’s Function Monte Carlo solutions up to A = 102000 - Ab-initio solutions for A = 12 via effective operators

Present day applications with effective operators:A = 16 solved with realistic NN interactionsA = 14 solved with realistic NN and NNN interactions

Constituent Quark Models of Exotic MesonsR. Lloyd, PhD Thesis, ISU 2003Phys. Rev. D 70: 014009 (2004)

H = T + V(OGE) + V(confinement)

Solve in HO basis as a bare H problem & study dependence on cutoff

Symmetries:Exact treatment of color degree of freedom <-- Major new accomplishmentTranslational invariance preservedAngular momentum and parity preserved

Next generation:More realistic H fit to wider range of mesons and baryonsSee preliminary results below.

Beyond that generation:Heff derived from QCD using light-front quantization

Nmax/2

Mas

s(M

eV)

Burning issues

Demonstrate degeneracy - Spontaneous Symmetry BreakingTopological features - soliton mass and profile (Kink, Kink-Antikink)

Quantum modes of kink excitationPhase transition - critical coupling, critical exponent and

the physics of symmetry restorationRole and proper treatment of the zero mode constraint

Chang’s Duality

4 in 1+1 Dimensions

4 in 1+1 DimensionsDLCQ with Coherent State Analysis

A. Derive the Hamiltonian and quantize it on the light front, investigate coherent state treatment of vacuum A. Harindranath and J.P. Vary, Phys Rev D36, 1141(1987)

B. Obtain vacuum energy as well as the mass and profile functions of soliton-like solutions in the symmetry-broken phase:

PBC: SSB observed, Kink + Antinkink ~ coherent state!Chakrabarti, Harindranath, Martinovic, Pivovarov and Vary,Phys. Letts. B to be published; hep-th/0310290.

APBC: SSB observed, Kink ~ coherent state!Chakrabarti, Harindranath, Martinovic and Vary, Phys. Letts. B582, 196 (2004); hep-th/0309263

C. Demonstrate onset of Kink Condensation:Chakrabarti, Harindranath and Vary, Phys. Rev. D71, 125012(2005); hep-th/0504094

€

Lagrangian density

=12

(∂μφ∂μφ + μ 2φ2 ) −λ4!

φ 4

Hamiltonian quantized in light front coordinates with, for example, anti - periodic boundary conditions:H = H 0 + H1 + H2

H0 = −μ 2 1n

an+

n =1/2

K

∑ an

H1 =λ4π

1N kl

21

Nmn2

ak+al

+aman

klmnk ≤l ,m ≤n

K

∑ δm + n,k +l

H2 =λ

4π1

N lmn2 [

ak+alaman + an

+am+ al

+ak

klmnk ,l ≤m ≤ n

K

∑ ]δ k,m + n+ l

N kl2 =1, k ≠ l N lmn

2 =1,l ≠ m ≠ n = 2!,k = l = 2!, l = m ≠ n,l ≠ m = n = 3!,l = m = nwhere all indecies are half odd integers.

L

€

nkk∑ k = K

H ψ i = E i ψ i

Fully covariant mass - squared spectra emergesM i

2 = KE i

Must extrapolate to the continuum limit, K - > ∞

At finite K, results are compared with results from a constrained variational treatment based on the coherent state ansatz of J.S. Rozowski and C.B. Thorn, Phys. Rev. Lett. 85, 1614 (2000).

Set up a normalized and symmetrized set of basis states, where, with representing the number of bosons with light front momentum k:

€

nk

DLCQ matrix dimension in even particle sector (APBC)

K Dimension16 33632 1421940 6724345 16549850 38925355 88096260 1928175

€

χ(n) = K an+an K

Normalized:

nn∑ χ (n) = K

Light cone momentum fraction:x = n / K

Light front momentum probability density in DLCQ:

Compares favorably with results from a constrainedvariational treatment based on the coherent state

€

M02 = E0vK + M kinkK

2

=> E0 = M02 / K = E0v + M kinkK

Note:φ → −φ symmetry=> Decoupling of even and odd boson sectors=> Exact degeneracy only in the K → ∞ limit

Extract the Vacuum energy density and Kink Mass

All results to date are in the broken phase:

€

μ 2 = −1

Define a Ratio = [M2even - M2

odd]/ [Vac Energy Dens]2

PBC

fit range

Vacuum Energy

Classical DLCQ-APBC

DLCQ-PBCzero modes?

0.5 -37.70 -37.81(7) -37.90(4)

1.0 -18.85 -18.71(5) -18.97(2)

1.25 -15.08 -14.91(5) 15.19(5)

Soliton Mass

Classical Semi-Classical

DLCQ-APBC

DLCQ-PBCzero modes?

0.5 11.31 10.84 11.6(2) 11.26(4)

1.00 5.657 5.186 5.22(8) 5.563(7)

1.25 4.526 4.054 4.07(6) 4.43(4)

Fourier Transform of the Soliton (Kink) Form Factor Ref: Goldstone and Jackiw

€

φc (x− − a) = dq+ exp −i2

q+a ⎧ ⎨ ⎩

⎫ ⎬ ⎭−∞

∞

∫ K + q+ Φ(x −) K

Evaluated within DLCQ choosing a = 0:

=14πl

l =12

,32

,52

,...

∞

∑ K ± l ale−lπ

x −

L + al+e

lπx −

L ⎧ ⎨ ⎩

⎫ ⎬ ⎭ K

Note: Issue of the relative phases - fix arbitrary phases viaguidance of the the coherent state analysis:

€

Sign K + l al+ K{ } = +

Sign K − l al K{ } = −

Cancellation of imaginary terms and vacuum expectation value, ,are non-trivial tests of resulting kink structure.

€

φ

Quantum kink (soliton) in scalar field theory at = 1

Can we observe a phase transition in ?

€

1+14

How does a phase transition develop as a function of increased coupling?

What are the observables associated with a phase transition?

What are its critical properties (coupling, exponent, …)?

Continuum limit of the critical coupling

Light front momentum distribution functions

What is the nature of this phase transition?

(1) Mass spectroscopy changes(2) Form factor character changes -> Kink condensation signal(3) Parton distribution changes

QCD applications in the -link approximation for mesons

€

qq + qq

S. Dalley and B. van de Sande, Phys. Rev. D67, 114507 (2003); hep-ph/0212086

D. Chakrabarti, A. Harindranath and J.P. Vary, Phys. Rev. D69, 034502 (2004); hep-ph/0309317

DLCQ for longitudinal modes and a transverse momentum lattice

Adopt QCD Hamiltonian of Bardeen, Pearson and Rabinovici Restrict the P-space to q-qbar and q-qbar-link configurations Does not follow the effective operator approach (yet) Introduce regulators as needed to obtain cutoff-independent spectra

Full q-qbar comps q-qbar-link comps

Lowest state

Fifth state

Conclusions

• Similarity of “two-scale” problems in quantum systems with many degrees of freedom• Ab-initio theory is a convergent exact method for solving many-particle Hamiltonians• Two-body cluster approximation (easiest) suitable for many observables• Method has been demonstrated as exact in the nuclear physics applications• Quasi-exact results for 1+1 scalar field theory obtained • Non-perturbative vacuum expectation value (order parameter)• Kink mass & profile obtained• Critical properties (coupling, exponent) emerging• Evidence of Kink condensation obtained• Sensitivity to boundary conditions needs further study• Role of zero modes yet to be fully clarified (Martinovic talk tomorrow)• Advent of low-cost parallel computing has made new physics domains accessible: algorithm improvements have achieved fully scalable and load-balanced codes.

Future Plans

Apply LF-transverse lattice and basis functions to qqq systems (Stan Brodsky’s talk)

Investigate alternatives Effective Operator methods (Marvin Weinstein’s - CORE)

Improve semi-analytical approaches for accelerating convergence (Non-perturbative renormalization approaches - for discussion)